Not sure why you wouldn't want to use essential oils as they are highly concentrated so a tiny amount goes a long way. They are also completely natural so unless you are allergic to the plant it comes from, you shouldn't have to worry about reactions.
You can buy cheaply perfumed things at any dollar store, but synthetic fragrances are full of things that are very bad for you. Some of the worst carcinogenic substances are hidden under the label "fragrance" many of them used in household cleaners and other products. Many have been linked to increased breast cancer and they are known endocrine disrupting substances.
To pull odors out you could sew up some coffee grounds into a sachet to throw in the drier. It won't scent your laundry but it does absorb any odors that might linger. You could put lavender buds into a sachet as well and dry your clothes with it - that would impart a mild scent most likely.
If you don't like essential oils - nothing beats hanging laundry outside on a nice, breezy day depending on where you live.