Will Carington, works for Noodle
To find a free or low-cost preschool program in your area, you have a few options. First, I'd recommend performing a preschool search here on Noodle. Once you have a list of programs, pick a few that interest you and contact them directly about tuition – many private preschools offer some form of financial assistance, so it's worth asking!
Your second option is to see what preschool programs are offered by your local school district. Roane County Schools does offer a pre-k program, however children must be four years old in order to enroll. To learn more, contact the Roane County Schools Pre-K Office at (865) 376-5592, Ext. 1107. At the very least, an administrator there might have ideas about what low-cost preschool options exist in your area for three-year-olds.
Finally, I'd also recommend that you look into Head Start, a federal program which provides free early childhood education services to families who qualify. To find a program near you, use the Head Start locator tool. Many of the Head Start programs in the Harriman, Tennessee area are run by the Mid-East Community Action Agency. For more information about their programs, locations, eligibility requirements, etc., call (865) 248-8661 or visit their website.
Hope this helps, and good luck with your search!