Well, if you have a bank account you can take checks. Or money orders.
But checks can bounce and money orders can be fake. Leaving you responsible for them.
To accept credit and debit cards without paypal you're going to need a business setup. You can get a reader for your phone and stuff, but you will pay a fee each time it is used. And I don't know how that would work for transactions online.
I've never had issues with paypal, nor has anyone I've talked to.
I've seen some people sell services via Etsy.. but I can't recall what payment method Etsy uses...
Like the list an 'item' and a price like "I'll make a dress for you in any size and design for this ammount of money". As people also sell art there you might want to look up if people can list commissions there. Like I dunno if someone noted you on dA and you agreed it would be 30 dollars you could list a service at that cost for that person only to buy.
Etsy uses your credit card. It applies selling fees and profits at the end of the month, minus the 35% cut for the site. So you don't want to agree on a price here then go through them, you'll lose a 3rd of your money.
They take paypal, credit card, and 'other' which I'll assume is electronic bank transfer kind of thing.