AMD claims that its CPU architecture is immune to the attacks, though this statement may have been made prematurely. Because exploits have different variants, current analyses only show that AMD CPUs are safe against the current version of Meltdown, which focuses primarily on Intel’s architecture. The Meltdown paper has the following statement:
"We also tried to reproduce the Meltdown bug on several ARM and AMD CPUs. However, we did not manage to successfully leak kernel memory with the attack described in Section 5, neither on ARM nor on AMD. The reasons for this can be manifold. First of all, our implementation might simply be too slow and a more optimized version might succeed. For instance, a more shallow out-of-order execution pipeline could tip the race condition towards against the data leakage. Similarly, if the processor lacks certain features, e.g., no re-order buffer, our current implementation might not be able to leak data. However, for both ARM and AMD, the toy example as described in Section 3 works reliably, indicating that out-of-order execution generally occurs and instructions past illegal memory accesses are also performed."
So far, then, AMD processors haven't been affected by Meltdown, but they're not immune against Spectre, which uses a wider range of speculative execution features. Spectre impacts CPUs from AMD, ARM, and Intel, and the cure may in some ways be worse than the disease, depending on how much it impairs performance.