We had the same issue in our Orlando kitchen after a few years. What finally worked for us was scrubbing the grout lines with a stiff brush and an oxygen-based cleaner, then sealing the grout once it was fully dry. Two years later it still looks bright, and we reseal about once a year now.

" Our kitchen backsplash has white grout that's starting to look dingy "

What type of grout is it? How old?

We had the same dingy-grout problem behind our stove. What worked best for me was a paste of baking soda and water with an old toothbrush, then sealing the grout once it was dry. The sealer made the biggest long-term difference, it keeps grease splatter from soaking in. Now I just wipe it down with regular soapy water when I clean the counters.