You are correct that basic genomic associations with Chi-square or linear regression procedures are not appropriate for Family-based studies and study populations with significant cryptic relatedness. Those types of data sets would inflate your test-statistics and result in many false-positive results. There are a few different techniques you can use based on the structure of your data set.
Family-based studies were phenotypes and genotypes of the parents and (hopefully multiple) offspring are known allow you to use the transmission disequilibrium test (TDT), family based association test (FBAT), and pedigree disequilibrium test (PDT) methods. I am less familiar with these methods and will not expand upon them.
For non-family-based studies with significant relatedness the predominant methods are based on mixed effect modeling. This approach utilizes a kinship matrix (pedigree, genetic, or hybrid) to prevent inflation. For case-control phenotypes, I would suggest using GEMMA (http://www.xzlab.org/software.html) or GCTA (http://cnsgenomics.com/software/gcta/#Overview), but of course their are numerous other programs out there (see manuscript below). Personally, I have used GCTA on a livestock data set (tremendous number of paternal half-siblings, cousins, and inbreeding) and it controlled for inflation phenomenally.
I hope this helped.
Article Comparison of Methods to Account for Relatedness in Genome-W...