Quantum gravity effects would appear under extreme conditions (where ordinary classical GR breaks down and develops "singularities") such as in connection with cosmological big bounce or black holes.
So most of the research papers studying ways to test are about BB or BH, and most have to do with either the CMB (ancient light from early universe, showing fluctuations in density) or with radiation that might be detected coming from the evaporation of black holes.
I can only toss out some links to give an idea of the kind of things being studied. I don't know of anybody who has the immediate direct ANSWER of how to test today's QG theories with today's technical means.
This is just to get the flavor. It is not something to try to understand in depth, all still preliminary and nowhere near ready for mass audience.
http://physics.republika.pl/PRD7.pdf (Physical Review D 2010)
Observing the big bounce with tensor modes in the cosmic microwave background:
Phenomenology and fundamental loop quantum cosmology parameters
Some slides for a conference talk by the same people--more visual and less texty:
http://indico.in2p3.fr/getFile.py/a...onId=13&resId=0&materialId=slides&confId=2779
According to one version of QG theory clouds of microscopic BH would exist and show characteristic radiation. The particular theory could be tested (and perhaps falsified) if one looks for that characteristic radiation (and perhaps does not find it.) Strange rather bold idea...FWIW here is the paper:
http://arxiv.org/abs/1202.0412
Emission spectra of self-dual black holes
Sabine Hossenfelder, Leonardo Modesto, Isabeau Prémont-Schwarz
(Submitted on 2 Feb 2012)
We calculate the particle spectra of evaporating self-dual black holes that are potential dark matter candidates. ... In this limit, we then derive the number-density of the primary emission particles,... We finally arrive at the expression for the spectrum of secondary particle emission from a dark matter halo constituted of self-dual black holes.
15 pages, 6 figures
Here's a more popularized account of some different proposals for testing Loop QG:
Physicists propose test for loop quantum gravity
www.physorg.com/pdf244805864.pdf