Are there any geological periods found on top of each other that have dinosaurs of that period with in them?
There are definitely many examples of overlying Mesozoic strata that contain characteristic dinosaur fossils (and ichnofossils – i.e. trace fossils). One of the best examples is the Mesozoic stratigraphy of Utah and Colorado. However, most geologic intervals are characterised by less ‘famous’ fossil material (e.g. molluscs, trilobites or pollen grains). This does not render correlations involving dinosaurs less valid. 

Consider a hypothetical case where some dinosaur footprints are found in a mudstone layer. The mudstone also contains distinctive grains of pollen. In the layer above the mudstone we see no more dinosaur footprints but we still have the same pollen grains. These pollen have been found in other strata (perhaps on the other side of the world!) while the dinosaur foot prints are much more rare. In this case we will be able to correlate these to rock layers much more widely using the pollen than from the dinosaur prints. However, if the mudstone layer is the youngest geological unit known to contain the fossil remnants of that particular dinosaur species (assuming it had a distinctive footprint) then we can assume that the species became extinct. Extinctions are very useful for correlating the relative age of geological units, therefore the last appearance of the dinosaur footprints will provide a tighter age control than the pollen (which persists in the rock record).

I should add that stratigraphic corrections based on the fossil record can be very complex. The scientists who study the fossil record (palaeontologists) spend decades trying to decipher the exact details of the evolution of life on this planet preserved in rocks. Due to the complexity of this science it is very easy for a layperson to get confused or make false assessments.