I know why DRAM is slower to write than to read, but why is the L1 & L2 cache RAM slower to write?
Are you really sure the caches are slower to write than to read?  I suspect what you heard was the overall operation, not just the SRAM read or write time.  Writing to a cache is a more expensive operation because eventually the slower memory backing the cache must be written to.  It also means a cache block can't simply be reused without flushing it first if it's been written to and that write hasn't been propagated back to the main memory.

The actual write operation is probably the same speed, but the overall process of writing, and all the stuff that ultimately puts in motion, is longer.