There is an order. It comes up about as often as alphabetical order comes up for us – which is, to be honest, not very.
The mnemonic I learnt from Wikipedia for remembering the order of the columns is “Ah, Kana Symbols. Take Note How Many You Remeber Well” (and then tack ん on the end, though since no word in the dictionary starts with ん, that’s not really something you have to remember). The rows, of course, go あ-い-う-え-お, which shouldn’t be too hard to remember.
If the dictionary I have to go by is indicative, then daikuten and glides don’t change the order – that is, び is sorted as though it’s a ひ, and しゃ is sorted as though it’s しや.