We use direct numerical simulation (DNS) based on spectral methods to simulate turbulent flow past rigid and flexible cylinders subject to vortex-induced vibrations (VIV). We present comparisons of amplitude, and lift and drag forces, at Reynolds number 1000 for a short and a long cylinder, and we examine differences between a traveling wave response and a standing wave response. The DNS data suggest that the often-used empirical formula proposed by Skop, Griffin & Ramberg in 1977 overpredicts the drag coefficient. We propose an appropriate modification and present preliminary results that indicate that low-dimensional modeling may be an accurate and efficient approach in predicting forces in VIV. Given the lack of any benchmark experiments in VIV currently, the DNS results presented here, both distributions as well as span- and time-averaged quantities, should be helpful to experimentalists and modelers.