We present results of an analysis of time series data observed in sunspot umbral regions. The data were obtained in the context of the SOHO Joint Observing Program (JOP) 97 in September 2000. This JOP included the Coronal Diagnostic Spectrometer (CDS) and the Michelson Doppler Imaging (MDI) instrument, both part of SOHO, the TRACE satellite and various ground based observatories. The data was analysed by using both Fourier and wavelet time series analysis techniques. We find that oscillations are present in the umbra at all temperatures investigated, from the temperature minimum as measured by TRACE 1700 Å up to the upper corona as measured by CDS Fe Xvi 335 Å (log T=6.4 K). Oscillations are found to be present with frequencies in the range of 5.4 mHz (185 s) to 8.9 mHz (112 s). Using the techniques of cross-spectral analysis time delays were found between low and high temperature emission suggesting the possibility of both upward and downward wave propagation. It is found that there is typically a good correlation between the oscillations measured at the different emission temperatures, once the time delays are taken into account. We find umbral oscillations both inside and outside of sunspot plume locations which indicates that umbral oscillations can be present irrespective of the presence of these sunspot plumes. We find that a number of oscillation frequencies can exist co-spatially and simultaneously i.e. for one pixel location three different frequencies at 5.40, 7.65 and 8.85 mHz were measured. We investigate the variation of the relative amplitudes of oscillation with temperature and find that there is a tendency for the amplitudes to reach a maximum at the temperature of O Iii (and less typically O V and Mg Ix) and then to decrease to reach a minimum at the temperature of Mg X (log T=6.0 K), before increasing again at the temperature of Fe Xvi. We discuss a number of possible theoretical scenarios that might explain these results. From a measurement of propagation speeds we suggest that the oscillations we observe are due to slow magnetoacoustic waves propagating up along the magnetic field lines.