A LinHES system can use most modern and "semi-modern" hardware.
Some users report a fully functional system driven by an antiquated AMD Athlon (800 MHz)/nforce2 CPU/MB.
For HD playback a more powerful CPU (Pentium4 2.8GHz or better) or a GPU capable of VDPAU such as an NVIDIA 8400GS, 9500GT, or GF 210.
For a complete table of NVIDIA cards and their VDPAU support, see this table.
Intel Atom-based PCs are also popular among LinHES users for their low heat output and ultra low power consumption.
Again, this low-power CPU needs to be paired with a VDPAU compatible GPU or else view HD content will not be possible. Most Atom MB/CPU combos are paired with one, for example, see offerings by Intel and Zotac.
LinHES can also run in a virtual machine (verified to work in Oracle's VirtualBox v3.x and 4.0) if would-be users would prefer to test drive it in a sandbox.