While Peter Humphries' answer is entirely correct for the question asked by the OP, the technique indicated there is far from addressing the most general situation.

The most basic technique towards this problem, given by Margulis' in his celebrated thesis for general Anosov flows, is what's known as the flow-box argument, which essentially means (in the case we have at hand) to thicken a period, and then push it along the geodesic flow (split-cartan) direction, as the unipotent direction is horospheric wrt the cartan action, it grows, while the cartan direction stays natural and the opposite unipotent direction gets shrunk (this is the property of the Anosov splitting), using this simple geometrical observation, together with quantitative mixing estimates for the Cartan action (which are merely quantification of the Howe-Moore theorem) gives you this result. The most basic mixing estimate is due to Harish-Chandra (say for $K$-finite vectors, which are the analogues of trigonometrical polynomials in this case), later it has been generalized to practically any smooth vectors in any Sobolov space (HC, Howe, Moore,Ratner), and the most general form (say for any Holder-cont. function) is due to Kleinbock-Margulis and Katok-Spatzier. The exact rate of mixing is obviously reflected in the spectral decomposition of your space (in the non arithmetic case you will need to use Selberg's (or maybe Langlands') results, which are more complicated), as Peter mentioned in his answer.
This approach is described nicely in Manfred's book.

Moreover, as this approach is holistic, it works for any lattice, and your question then transformed into the question of estimating the spectral gap (more correctly, property $(\tau)$) for a set of lattices uniformly, namely if this set of lattices is a family of expanders, and nowadays we know quite a few of those (for example, Selberg's theorem tells you that for principal congruence subgroups).

For $SL_2 (\mathbb{Z})$ and in theory for any principal congruence subgroup, you can do better, by specific analysis of the constant term of the Eisenstein series, and this was first done by Peter Sarnak in one of his first articles, if I recall correctly he shows there that for $SL_{2}(\mathbb{Z})$ $\delta=3/4-\epsilon$ is equivalent to RH (probably you get worse epsilon and need GRH to handle the principal congruence case, but it should work, this relies on explicit description of Eisenstein series in the end).
This approach takes you further than Margulis' argument I've mentioned before, but it has limitations (can handle only principal congruence, very computational heavy, Margulis' method can be bootstrapped to quantify the Furstenberg/Dani-Smillie equidistribution theorem where this spectral approach is quite limited towards this problem).

Burger (and later Strombergsson, Flaminio-Forni) found a different approach, which relies much less on the artihmetics and gives you great results for practically any lattice (this deal with the equidistribution result of Furstenberg (Burger) and Dani-Smillie (Strombergsson), but it is not hard to see that the period theorem follows from this theorem).

Peter Humphries mentioned thin subgroups (infinite index) in his answer, so to make things clear, those are not lattices, and their spectral theory is different (and more complicated) due to Patterson-Sullivan and Lax-Phillips, and we do know many of expander type results for families there due to recent developments in arithmetic combinatorics (say the Bourgain-Gamburd technique). The problem there is to define the equidistirbution problem correctly, namely to which measure (as it is of infinite volume, the constants are not integrable and cannot serve as a test function, and one needs to consider different ways to average, for example consider Hopf's ergodic theorem instead of the regular one, or change to the Burger-Roblin measure, etc.) I will refer you to the magnificent articles of Oh-Mohammadi and Oh-Winter to see the results in those cases.