I'm planning to build a low-power storage server with many drives.
Can I put an Atom based board into this machine? Or even ARM? Will a weak CPU cut into my throughput? What about RAM?
I am planning to run Linux and lvm.
There's a few factors it's going to depend on.
A hardware RAID controller to offload all the storage duties will help lower CPU usage, particularly for anything more demanding than RAID-5. That said, 10 disks is quite a few for a single array -- you'll probably want either RAID-6 or (preferably) a hybrid RAID level (e.g. RAID-50).
However, if you want to present JBOD and use a higher layer storage mechanism (btrfs, mdraid, ZFS), then you might need to invest in a more powerful CPU. ZFS, in particular, can require a powerful CPU and gobs of RAM (and a decent SSD for L2ARC) if you want to take advantage of its higher end features or get maximum performance. You've mentioned Linux, so I'm assuming you don't intend to use ZFS.
10 disks use a lot of power, particularly 3.5" high-capacity disk -- you may find that the CPU doesn't really use that much power in the overall system. At least, if we're talking about modern, power efficient CPUs. If you're building a storage server with a 3.4 GHz Pentium 4 (Prescott) CPU, that's a whole different ball of wax.
Now, if it's going to be storage only and the CPU's not going to be burdened with dual-parity calculations, you should be able to get away with an Atom CPU. You could even use an ARM-based system, but unless you're buying a turnkey storage server, you'll probably be in for a lot of hassle getting everything up and running.
All that said, you'll probably find that a "decent" CPU (say, a relatively low end Intel Core-series/Pentium G or AMD Phenom/Fusion) will give you much more performance headroom without much extra up-front cost. It gives you the flexibility to do a lot more with the box than just serve files, and the idle power usage won't be significantly greater than most Atom CPUs.
I'd recommend using AMD Fusion E-350/E-450 but yes, even an atom based board would work just fine. A weak CPU should not have a major impact on your throughput. RAM may have some impact depending on the type of filesystem you choose. I'm not too knowledgeble when it comes to LVM, but a quick glance here seems to show that for LVM, performance is not tied to CPU or RAM. Now, CPU bit-width (32-bit/64-bit) does limit you in terms of LV size. For 32-bit CPU, it's 16TB and for 64-bit it's 8EB (which is more than enough).