We study a class of loop models, parameterized by a continuously varying loop fugacity n, on the hydrogen peroxide lattice, which is a three-dimensional cubic lattice of coordination number 3. For integer n>0, these loop models provide graphical representations for n-vector models on the same lattice, while for n=0 they reduce to the self-avoiding walk problem. We use worm algorithms to perform Monte Carlo studies of the loop model for n=0, 0.5, 1, 1.5, 2, 3, 4, 5 and 10 and obtain the critical points and a number of critical exponents, including the thermal exponent yt, magnetic exponent yh, and loop exponent yl. For integer n, the estimated values of yt and yh are found to agree with existing estimates for the three-dimensional O(n) universality class. The efficiency of the worm algorithms is reflected by the small value of the dynamic exponent z, determined from our analysis of the integrated autocorrelation times.