Ds1's tutor asks for cash also, but will accept a cheque for larger amounts. I pay her half-termly by cheque which is fine, but for anyone wanting to pay weekly, it's cash only. By paying several weeks at a time, I also get a discount of £1 a week.
I think the reason is to keep costs down. Cashing cheques will cost her as she is running a business, so it makes sense not to accept smaller amounts this way as she'll pay the same bank charge for a £5 cheque as she would for a £50 one iyswim. We have the same at Karate as well.