I am personaly running debian jessie with intel graphics, and I am very happy with the performance. I have in the past had issues with newer hardware and older kernels. my system is many moons old, so the drivers are pretty mature. I really dislike the nvidia drivers and have not recently experienced the ati. my lspci -vv output for my card is,
VGA compatible controller: Intel Corporation Mobile 4 Series Chipset Integrated Graphics Controller (rev 07)
...
Kernel driver in use: i915
a quick search for your board showed up
"The GigaByte GA-H87M-HD3 (rev. 1.x) does not support onboard graphics. Any system build that uses this motherboard therefore requires a separate graphics card, or a processor that has a GPU on the same die, such as AMD APU processors. There are 1 PCIe x16 slots on this motherboard. This means it is perfectly capable of accommodating the latest graphics cards, although it is important to try and use a graphics card with the same graphics card interface of PCIe v3.0, as anything below will not reach the motherboard's potential, and anything above will have its performance slashed to the bandwidth maximum of the GigaByte GA-H87M-HD3 (rev. 1.x)'s PCIe v3.0. The GigaByte GA-H87M-HD3 (rev. 1.x) does not support multiple graphics cards via Nvidia SLI or AMD Crossfire. "
good luck.
VGA compatible controller: Intel Corporation Mobile 4 Series Chipset Integrated Graphics Controller (rev 07)
...
Kernel driver in use: i915
a quick search for your board showed up
"The GigaByte GA-H87M-HD3 (rev. 1.x) does not support onboard graphics. Any system build that uses this motherboard therefore requires a separate graphics card, or a processor that has a GPU on the same die, such as AMD APU processors. There are 1 PCIe x16 slots on this motherboard. This means it is perfectly capable of accommodating the latest graphics cards, although it is important to try and use a graphics card with the same graphics card interface of PCIe v3.0, as anything below will not reach the motherboard's potential, and anything above will have its performance slashed to the bandwidth maximum of the GigaByte GA-H87M-HD3 (rev. 1.x)'s PCIe v3.0. The GigaByte GA-H87M-HD3 (rev. 1.x) does not support multiple graphics cards via Nvidia SLI or AMD Crossfire. "
good luck.