We carry out a direct search for bar-like non-circular flows in intermediate-inclination, gas-rich disk galaxies with a range of morphological types and photometric bar classifications from the first data release (DR1) of the CALIFA survey. We use the DiskFit algorithm to apply rotation only and bisymmetric flow models to H$α$ velocity fields for 49/100 CALIFA DR1 systems that meet our selection criteria. We find satisfactory fits for a final sample of 37 systems. DiskFit is sensitive to the radial or tangential components of a bar-like flow with amplitudes greater than $15\,$km$\,$s$^{-1}$ across at least two independent radial bins in the fit, or ~2.25 kpc at the characteristic final sample distance of ~75 Mpc. The velocity fields of 25/37 $(67.6^{+6.6}_{-8.5}\%)$ galaxies are best characterized by pure rotation, although only 17/25 $(68.0^{+7.7}_{-10.4}\%)$ of them have sufficient H$α$ emission near the galaxy centre to afford a search for non-circular flows. We detect non-circular flows in the remaining 12/37 $(32.4^{+8.5}_{-6.6}\%)$ galaxies. We conclude that the non-circular flows detected in 11/12 $(91.7^{+2.8}_{-14.9}\%)$ systems stem from bars. Galaxies with intermediate (AB) bars are largely undetected, and our detection thresholds therefore represent upper limits to the amplitude of the non-circular flows therein. We find 2/23 $(8.7^{+9.6}_{-2.9}\%)$ galaxies that show non-circular motions consistent with a bar-like flow, yet no photometric bar is evident. This suggests that in ~10% of galaxies either the existence of a bar may be missed completely in photometry or other processes may drive bar-like flows and thus secular galaxy evolution. ; Comment: Accepted for publication in MNRAS (16 pages, 10 figures, 3 tables). Online Only Appendices: http://tinyurl.com/pav3cdx. High Resolution Figures: http://tinyurl.com/pr824uu