We present efficient cache-oblivious algorithms for some well-studied string problems in bioinformatics including the longest common subsequence, global pairwise sequence alignment and three-way sequence alignment (or median), both with affine gap costs, and RNA secondary structure prediction with simple pseudoknots. For each of these problems, we present cache-oblivious algorithms that match the best-known time complexity, match or improve the best-known space complexity, and improve significantly over the cache-efficiency of earlier algorithms. We present experimental results which show that our cache-oblivious algorithms run faster than software and implementations based on previous best algorithms for these problems.