  Context. Establishing the multiplicity of O-type stars is the first step
towards accurately determining their stellar parameters. Moreover, the
distribution of the orbital parameters provides observational clues to the way
that O-type stars form and to the interactions during their evolution. Aims.
Our objective is to constrain the multiplicity of a sample of O-type stars
belonging to poorly investigated OB associations in the Cygnus complex and for
the first time to provide orbital parameters for binaries identified in our
sample. Such information is relevant to addressing the issue of the binarity in
the context of O-type star formation scenarios. Methods. We performed a
long-term spectroscopic survey of nineteen O-type stars. We searched for radial
velocity variations to unveil binaries on timescales from a few days up to a
few years, on the basis of a large set of optical spectra. Results. We confirm
the binarity for four objects: HD193443, HD228989, HD229234 and HD194649. We
derive for the first time the orbital solutions of three systems, and we
confirm the values of the fourth, showing that these four systems all have
orbital periods shorter than 10 days. Besides these results, we also detect
several objects that show non-periodic line profile variations in some of their
spectral lines. These variations mainly occur in the spectral lines, that are
generally affected by the stellar wind and are not likely to be related to
binarity. Conclusions. The minimal binary fraction in our sample is estimated
to be 21%, but it varies from one OB association to the next. Indeed, 33% of O
stars belonging to CygOB1 are binary systems, 0% in CygOB3, 0% in CygOB8, and
33% in CygOB9. Our spectroscopic investigation also stresses the absence of
long-period systems among the stars in our sample...
