We present first results from radio observations with the Murchison Widefield Array seeking to constrain the power spectrum of 21 cm brightness temperature fluctuations between the redshifts of 11.6 and 17.9 ( 113 and 75 MHz ) . Three hours of observations were conducted over two nights with significantly different levels of ionospheric activity . We use these data to assess the impact of systematic errors at low frequency , including the ionosphere and radio-frequency interference , on a power spectrum measurement . We find that after the 1-3 hours of integration presented here , our measurements at the Murchison Radio Observatory are not limited by RFI , even within the FM band , and that the ionosphere does not appear to affect the level of power in the modes that we expect to be sensitive to cosmology . Power spectrum detections , inconsistent with noise , due to fine spectral structure imprinted on the foregrounds by reflections in the signal-chain , occupy the spatial Fourier modes where we would otherwise be most sensitive to the cosmological signal . We are able to reduce this contamination using calibration solutions derived from autocorrelations so that we achieve an sensitivity of 10 ^ { 4 } mK on comoving scales k \lesssim 0.5 h Mpc ^ { -1 } . This represents the first upper limits on the 21 cm power spectrum fluctuations at redshifts 12 \lesssim z \lesssim 18 but is still limited by calibration systematics . While calibration improvements may allow us to further remove this contamination , our results emphasize that future experiments should consider carefully the existence of and their ability to calibrate out any spectral structure within the EoR window .