Works on processors going back as far as 2007 (the oldest I have access to now is an Athlon 64 X2 6000+), but the example code relies on an instruction that the Atom D510 does not suport.
Because Spectre seems to be an intrinsic problem with out-of-order execution, which is almost as old as the FDIV bug in intel processors, I would be very surprised if the Atom D510 did not turn out to be susceptible using other methods as outlined in the paper.
EDIT: I originally suspected this instruction was CLFLUSH and erroneously claimed the D510 doesn't support sse2. It does support sse2, so it must be that it does not support the RDTSCP instruction used for timing.
EDIT: This gets very interesting. I made some modifications to use a CPUID followed by RDTSC, which now runs without illegal instructions and works everywhere the previous version worked. Except on the D510, this runs but I cannot get the leak to happen despite exploring values of CACHE_HIT_THRESHOLD. Could the Atom D510 really be immune from Spectre?