I was able to flash the MSI H81M-U33 with flashrom. The built-in M-Flash wasnt working. Flashed from 6.0 to 6.6.

BIOS Information
        Vendor: American Megatrends Inc.
        Version: V6.6
        Release Date: 07/22/2014
        Address: 0xF0000
        Runtime Size: 64 kB
        ROM Size: 8192 kB

Base Board Information
        Manufacturer: MSI
        Product Name: H81M-E33 (MS-7817)

Linux Zaphod 3.13.0-38-generic #65-Ubuntu SMP Thu Oct 9 11:36:50 UTC 2014 x86_64 x86_64 x86_64 GNU/Linux

processor    : 0
vendor_id    : GenuineIntel
cpu family    : 6
model        : 60
model name    : Intel(R) Pentium(R) CPU G3220 @ 3.00GHz
stepping    : 3
microcode    : 0x1a

root@Zaphod:~# flashrom -V -p internal -r BIOS.good
flashrom v0.9.6.1-r1563 on Linux 3.13.0-38-generic (x86_64)
flashrom is free software, get the source code at http://www.flashrom.org

flashrom was built with libpci 3.1.9, GCC 4.7.1, little endian
Command line (5 args): flashrom -V -p internal -r BIOS.good
Calibrating delay loop... OS timer resolution is 1 usecs, 2998M loops per second, 10 myus = 10 us, 100 myus = 107 us, 1000 myus = 999 us, 10000 myus = 9997 us, 4 myus = 4 us, OK.
Initializing internal programmer
No coreboot table found.
DMI string system-manufacturer: "MSI"
DMI string system-product-name: "MS-7817"
DMI string system-version: "1.0"
DMI string baseboard-manufacturer: "MSI"
DMI string baseboard-product-name: "H81M-E33 (MS-7817)"
DMI string baseboard-version: "1.0"
DMI string chassis-type: "Desktop"
Found chipset "Intel Lynx Point" with PCI ID 8086:8c5c.
This chipset is marked as untested. If you are using an up-to-date version
of flashrom *and* were (not) able to successfully update your firmware with it,
then please email a report to flashrom@flashrom.org including a verbose (-V) log.
Thank you!
Enabling flash write... enable_flash_ich_dc_spi: unknown ICH generation. Please report!

0xfff80000/0xffb80000 FWH IDSEL: 0x0
0xfff00000/0xffb00000 FWH IDSEL: 0x0
0xffe80000/0xffa80000 FWH IDSEL: 0x1
0xffe00000/0xffa00000 FWH IDSEL: 0x1
0xffd80000/0xff980000 FWH IDSEL: 0x2
0xffd00000/0xff900000 FWH IDSEL: 0x2
0xffc80000/0xff880000 FWH IDSEL: 0x3
0xffc00000/0xff800000 FWH IDSEL: 0x3
0xff700000/0xff300000 FWH IDSEL: 0x4
0xff600000/0xff200000 FWH IDSEL: 0x5
0xff500000/0xff100000 FWH IDSEL: 0x6
0xff400000/0xff000000 FWH IDSEL: 0x7
0xfff80000/0xffb80000 FWH decode enabled
0xfff00000/0xffb00000 FWH decode enabled
0xffe80000/0xffa80000 FWH decode enabled
0xffe00000/0xffa00000 FWH decode enabled
0xffd80000/0xff980000 FWH decode enabled
0xffd00000/0xff900000 FWH decode enabled
0xffc80000/0xff880000 FWH decode enabled
0xffc00000/0xff800000 FWH decode enabled
0xff700000/0xff300000 FWH decode enabled
0xff600000/0xff200000 FWH decode enabled
0xff500000/0xff100000 FWH decode enabled
0xff400000/0xff000000 FWH decode enabled
Maximum FWH chip size: 0x100000 bytes
BIOS Lock Enable: disabled, BIOS Write Enable: disabled, BIOS_CNTL is 0x8
Root Complex Register Block address = 0xfed1c000
GCS = 0xc61: BIOS Interface Lock-Down: enabled, Boot BIOS Straps: 0x3 (unknown)
Top Swap : not enabled
SPIBAR = 0xfed1c000 + 0x3800
0x04: 0xf008 (HSFS)
HSFS: FDONE=0, FCERR=0, AEL=0, BERASE=1, SCIP=0, FDOPSS=1, FDV=1, FLOCKDN=1
WARNING: SPI Configuration Lockdown activated.
Reading OPCODES... done
0x06: 0x3f00 (HSFC)
HSFC: FGO=0, FCYCLE=0, FDBC=63, SME=0
0x08: 0x007fffc0 (FADDR)
0x50: 0x0000ffff (FRAP)
BMWAG 0x00, BMRAG 0x00, BRWA 0xff, BRRA 0xff
0x54: 0x00000000 FREG0: Flash Descriptor region (0x00000000-0x00000fff) is read-write.
0x58: 0x07ff0200 FREG1: BIOS region (0x00200000-0x007fffff) is read-write.
0x5C: 0x01ff0001 FREG2: Management Engine region (0x00001000-0x001fffff) is read-write.
0x90: 0xc0 (SSFS)
SSFS: SCIP=0, FDONE=0, FCERR=0, AEL=0
0x91: 0xfc4010 (SSFC)
SSFC: SCGO=0, ACS=0, SPOP=0, COP=1, DBC=0, SME=0, SCF=4
0x94: 0x0006     (PREOP)
0x96: 0x043b     (OPTYPE)
0x98: 0x05203b02 (OPMENU)
0x9C: 0x0000019f (OPMENU+4)
0xA0: 0x00000000 (BBAR)
0xC4: 0x80802045 (LVSCC)
LVSCC: BES=0x1, WG=1, WSR=0, WEWS=0, EO=0x20, VCL=1
0xC8: 0x00002045 (UVSCC)
UVSCC: BES=0x1, WG=1, WSR=0, WEWS=0, EO=0x20, VCL=0
0xD0: 0x50444653 (FPB)

Enabling hardware sequencing because some important opcode is locked.
SPI Read Configuration: prefetching enabled, caching enabled, OK.
The following protocols are supported: FWH, Programmer-specific.
Probing for Programmer Opaque flash chip, 0 kB: Found 1 attached SPI flash chip with a density of 8192 kB.
The flash address space (0x000000 - 0x1acfff) is divided at address 0x653000 in two partitions.
The first partition ranges from 0x000000 to 0x652fff.
In that range are 1619 erase blocks with 4096 B each.
The second partition ranges from 0x653000 to 0x1acfff.
In that range are 429 erase blocks with 4096 B each.
Found Programmer flash chip "Opaque flash chip" (8192 kB, Programmer-specific) at physical address 0x0.
Probing for Atmel AT49LH002, 256 kB: probe_82802ab: id1 0xff, id2 0xfc, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for Intel 82802AB, 512 kB: probe_82802ab: id1 0x88, id2 0x6f, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for Intel 82802AC, 1024 kB: probe_82802ab: id1 0xff, id2 0xff, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for PMC Pm49FL002, 256 kB: probe_jedec_common: id1 0xff, id2 0xfc, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for PMC Pm49FL004, 512 kB: probe_jedec_common: id1 0x88, id2 0x6f, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for Sharp LHF00L04, 1024 kB: probe_82802ab: id1 0xff, id2 0xff, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for SST SST49LF002A/B, 256 kB: probe_jedec_common: id1 0xff, id2 0xfc, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for SST SST49LF003A/B, 384 kB: probe_jedec_common: id1 0x02, id2 0x00, id1 is normal flash content, id2 is normal flash content
Probing for SST SST49LF004A/B, 512 kB: probe_jedec_common: id1 0x88, id2 0x6f, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for SST SST49LF004C, 512 kB: probe_82802ab: id1 0x88, id2 0x6f, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for SST SST49LF008A, 1024 kB: probe_jedec_common: id1 0xff, id2 0xff, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for SST SST49LF008C, 1024 kB: probe_82802ab: id1 0xff, id2 0xff, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for SST SST49LF016C, 2048 kB: Chip size 2048 kB is bigger than supported size 1024 kB of chipset/board/programmer for FWH interface, probe/read/erase/write may fail. probe_82802ab: id1 0xb4, id2 0x73, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for ST M50FLW040A, 512 kB: probe_82802ab: id1 0x88, id2 0x6f, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for ST M50FLW040B, 512 kB: probe_82802ab: id1 0x88, id2 0x6f, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for ST M50FLW080A, 1024 kB: probe_82802ab: id1 0xff, id2 0xff, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for ST M50FLW080B, 1024 kB: probe_82802ab: id1 0xff, id2 0xff, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for ST M50FW002, 256 kB: probe_82802ab: id1 0xff, id2 0xfc, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for ST M50FW016, 2048 kB: Chip size 2048 kB is bigger than supported size 1024 kB of chipset/board/programmer for FWH interface, probe/read/erase/write may fail. probe_82802ab: id1 0xb4, id2 0x73, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for ST M50FW040, 512 kB: probe_82802ab: id1 0x88, id2 0x6f, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for ST M50FW080, 1024 kB: probe_82802ab: id1 0xff, id2 0xff, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for Winbond W39V040FA, 512 kB: probe_jedec_common: id1 0x88, id2 0x6f, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for Winbond W39V040FB, 512 kB: probe_jedec_common: id1 0x88, id2 0x6f, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for Winbond W39V040FC, 512 kB: probe_jedec_common: id1 0x88, id2 0x6f, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for Winbond W49V002FA, 256 kB: probe_jedec_common: id1 0xff, id2 0xfc, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for Winbond W39V080FA, 1024 kB: probe_jedec_common: id1 0xff, id2 0xff, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Probing for Winbond W39V080FA (dual mode), 512 kB: probe_jedec_common: id1 0x88, id2 0x6f, id1 parity violation, id1 is normal flash content, id2 is normal flash content
Found Programmer flash chip "Opaque flash chip" (8192 kB, Programmer-specific).
Reading flash... Reading 8388608 bytes starting at 0x000000.
done.
Restoring MMIO space at 0x7f74dc5858a0
Restoring PCI config space for 00:1f:0 reg 0xdc