The bitcoin world used to use random k values, but has largely thought better of it, because you need a way to make random numbers, and this is prone to implementation or platform bugs like the Android random number generation turning out to suck, or pulling a number from an HTTP URL at random.org and ending up using something like "HTTP is now disabled. Please use HTTPS".
The solution, which is also mentioned on the linked Wikipedia page, is to use a deterministic number instead of a random one. The number is based partly on the message that's being signed, so you can be confident that you won't deterministically generate the same value twice for different messages. This is what pretty much all crypto-currency wallet software does nowadays, and if they don't it's a red flag.
See this answer for more discussion of the pros and cons of this (basically all pros) including some comments by Pieter Wuille, who wrote libsecp256k1, the ECDSA library that is used in both Bitcoin and Ethereum.
Note that although you can do the signing without a random number, you still need random numbers to generate the keys in the first place, and that may be a good place to make you move your mouse around or whatever.