Megavoltage X-ray sources are commonly used for therapy planning, and knowledge of their spectral distribution is important for accurate dose calculations. There are many methods that could provide reasonable estimations of Megavoltage X-ray spectra, when very accurate attenuation data or at least very good set of initial guesses of the spectra are available. We present here a novel method, which can be used for accurate Megavoltage spectral reconstruction without any prior knowledge of spectral distribution; the method performs well even when the available transmission data are affected by noise. The method is based on a search for a smooth function that minimizes the differences between measured and calculated attenuation data. The algorithm is compared with well-known existing algorithms, using computer simulated data, both error-free and containing added random Gaussian noise. The reconstructed spectra are subsequently used to calculate the transmission through 50 cm of bone, muscle or fat tissue. It is shown that the relative errors in dose calculations, using the spectra reconstructed via this method, are significantly smaller than those obtained via well-established reconstruction algorithms--Truncated Singular Value Decomposition (TSVD) and Expectation Maximization (EM). These results suggest that the novel algorithm might be practical for routine Megavoltage therapy X-ray source calibration.