We present intermediate results from a long-term program of mapping the neutral absorption characteristics of the local interstellar medium, motivated by the availability of accurate and consistent parallaxes from the Hipparcos satellite. Equivalent widths of the interstellar NaI D-line doublet at 5890Å are presented for the lines-of-sight towards some 311 new target stars lying within ∼350pc of the Sun. Using these data, together with NaI absorption measurements towards a further ∼240 nearby targets published in the literature (for many of them, in the directions of molecular clouds), and the ∼450 lines-of-sight already presented by (Sfeir et al., 1999A&A...346..785S
), we show 3D absorption maps of the local distribution of neutral gas towards 1005 sight-lines with Hipparcos distances as viewed from a variety of different galactic projections. The data are synthesized by means of two complementary methods, (i) by mapping of iso-equivalent width contours, and (ii) by density distribution calculation from the inversion of column-densities, a method devised by Vergely et al. (2001A&A...366.1016V
). Our present data confirms the view that the local cavity is deficient in cold and neutral interstellar gas. The closest dense and cold gas ``wall'', in the first quadrant, is at ∼55-60pc. There are a few isolated clouds at closer distance, if the detected absorption is not produced by circumstellar material. The maps reveal narrow or wide ``interstellar tunnels'' which connect the Local Bubble to surrounding cavities, as predicted by the model of Cox & Smith (1974ApJ...189L.105C
). In particular, one of these tunnels, defined by stars at 300 to 600pc from the Sun showing negligible sodium absorption, connects the well known CMa void (Gry et al., 1985ApJ...296..593G
g), which is part of the Local Bubble, with the supershell GSH 238+00+09 (Heiles, 1998ApJ...498..689H
). High latitude lines-of-sight with the smallest absorption are found in two ``chimneys'', whose directions are perpendicular to the Gould belt plane. The maps show that the Local Bubble is ``squeezed'' by surrounding shells in a complicated pattern and suggest that its pressure is smaller than in those expanding regions. We discuss the locations of several HI and molecular clouds. Using comparisons between NaI and HI or CO velocities, in some cases we are able to improve the constraints on their distances. According to the velocity criteria, MBM 33-37, MBM 16-18, UT 3-7, and MBM 54-55 are closer than ∼100pc, and MBM 40 is closer than 80pc. Dense HI clouds are seen at less than 90pc and 85pc in the directions of the MBM 12 and MBM 41-43 clouds respectively, but the molecular clouds themselves may be far beyond. The above closest molecular clouds are located at the neutral boundary of the Bubble. Only one translucent cloud, G192-67, is clearly embedded within the LB and well isolated. These maps of the distribution of local neutral interstellar NaI gas are also briefly compared with the distribution of both interstellar dust and neutral HI gas within 300pc.