Not anymore. Bitcoin now requires dedicated hardware (ASICs). Other coins were designed to make use of ASICs impossible or impractical, requiring GPUs, but those still require a CPU to drive them.
New developments, such as Ethereum moving away from proof of work to proof of stake made GPUs unnecessary, but you still need a computer with a CPU to validate the blocks on the block-chain.
Edit: Even with ASICs mining bitcoin, you still need servers to distribute the work to them.
you have different methods of calculations for different types of crypto. most use gpu calculations, some may favor cpu calculations. there are a handful that do so using hard drives for instance.